How do you evaluate #|x|/x# as x approaches 0?

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Answer 1 · 2016-05-06T13:30:11

The limit does not exist.

I assume that the question is "How do you evaluate the limit of #absx/x# as #xrarr0#?"

Recall:

#absx = {(x,"if",x >= 0),(-x,"if",x < 0):}#.

So,

#absx/x = {(x/x,"if",x > 0),(-x/x,"if",x < 0):}#

# = {(1,"if",x > 0),(-1,"if",x < 0):}#.

Therefore,

#lim_(xrarr0^-)absx/x = -1# and #lim_(xrarr0^+)absx/x = 1#

Because the two one sided limits are not equal, the limit #lim_(xrarr0)absx/x# does not exist.